No dispersed car camping allowed inside the National Park boundary. The best, free, primitive camping option is on National Forest roads (accessible my most passenger cars).

BIG MEADOWS ROAD:
Get a good map and try Big Meadow Road #14S11, off Sequoia Highway 198.
This section of Sequoia National Forest is located in between both National Parks (Kings Canyon & Sequoia). The National Forest is now considered the Giant Sequoia National Monument.

KINGS RIVER CAMPING:
A very rough dirt road (Road #12S01) leads from Kings Hwy 180, down to the lower King River, where there are numerous small campgrounds and primitive camping options on both sides of the river. This road requires high clearance, and possibly 4WD during wet/snowy weather.

Kings Canyon Hiking

Grizzly Falls &Roaring River Falls

From Fresno take Hwy. 180, approximately 50 miles east to the
Big Stump Entrance Station. Take 180 to the left (north). Passing
thru a few Sequoia groves, continue until you enter King Canyon
NP. The South Branch of King River will be visible.

Grizzly Falls will be on your left & marked with a parking
area.
Continue on Hwy. 180 about 7 more miles to the parking area for
Roaring River Falls with a short, well-maintained trail to falls.